TY - GEN A1 - Stefanelli, Ulisse A1 - Wachsmuth, Daniel A1 - Wachsmuth, Gerd T1 - Optimal control of a rate-independent evolution equation via viscous regularization T2 - Discrete and Continuous Dynamical Systems - Series S (DCDS-S) N2 - We study the optimal control of a rate-independent system that is driven by a convex quadratic energy. Since the associated solution mapping is non-smooth, the analysis of such control problems is challenging. In order to derive optimality conditions, we study the regularization of the problem via a smoothing of the dissipation potential and via the addition of some viscosity. The resulting regularized optimal control problem is analyzed. By driving the regularization parameter to zero, we obtain a necessary optimality condition for the original, non-smooth problem. KW - Rate-independent system KW - optimal control KW - necessary optimality conditions Y1 - 2017 U6 - https://doi.org/10.3934/dcdss.2017076 SN - 1937-1632 SN - 1937-1179 VL - 10 IS - 6 SP - 1467 EP - 1485 ER - TY - GEN A1 - Wachsmuth, Daniel A1 - Wachsmuth, Gerd T1 - Regularization error estimates and discrepancy principle for optimal control problems with inequality constraints T2 - Control and Cybernetics N2 - In this article we study the regularization of optimization problems by Tikhonov regularization. The optimization problems are subject to pointwise inequality constraints in L²(Ω). We derive a-priori regularization error estimates if the regularization parameter as well as the noise level tend to zero. We rely on an assumption that is a combination of a source condition and of a structural assumption on the active sets. Moreover, we introduce a strategy to choose the regularization parameter in dependence of the noise level. We prove convergence of this parameter choice rule with optimal order. KW - source condition KW - discrepancy principle KW - non-smooth optimization KW - convex constraints KW - sparsity KW - regularization error estimates Y1 - 2011 SN - 0324-8569 VL - 40 IS - 4 SP - 1125 EP - 1158 ER - TY - GEN A1 - Wachsmuth, Gerd A1 - Wachsmuth, Daniel T1 - Convergence and Regularization Results for Optimal Control Problems with Sparsity Functional T2 - ESAIM: Control, Optimisation and Calculus of Variations N2 - Optimization problems with convex but non-smooth cost functional subject to an elliptic partial differential equation are considered. The non-smoothness arises from a L1-norm in the objective functional. The problem is regularized to permit the use of the semi-smooth Newton method. Error estimates with respect to the regularization parameter are provided. Moreover, finite element approximations are studied. A-priori as well as a-posteriori error estimates are developed and confirmed by numerical experiments. KW - Non-smooth optimization KW - sparsity KW - regularization error estimates KW - finite elements KW - discretization error estimates Y1 - 2011 U6 - https://doi.org/10.1051/cocv/2010027 SN - 1292-8119 SN - 1262-3377 VL - 17 IS - 3 SP - 858 EP - 886 ER - TY - GEN A1 - Geiger, Tobias A1 - Wachsmuth, Daniel A1 - Wachsmuth, Gerd T1 - Optimal control of ODEs with state suprema T2 - Mathematical Control and Related Fields N2 - We consider the optimal control of a differential equation that involves the suprema of the state over some part of the history. In many applications, this non-smooth functional dependence is crucial for the successful modeling of real-world phenomena. We prove the existence of solutions and show that related problems may not possess optimal controls. Due to the non-smoothness in the state equation, we cannot obtain optimality conditions via standard theory. Therefore, we regularize the problem via a LogIntExp functional which generalizes the well-known LogSumExp. By passing to the limit with the regularization, we obtain an optimality system for the original problem. The theory is illustrated by some numerical experiments. KW - Functional differential equations KW - differential equations with state suprema KW - optimality conditions KW - maximum principle KW - LogIntExp Y1 - 2021 U6 - https://doi.org/10.3934/mcrf.2021012 SN - 2156-8472 SN - 2156-8499 VL - 11 IS - 3 SP - 555 EP - 578 ER - TY - GEN A1 - Wachsmuth, Daniel A1 - Wachsmuth, Gerd T1 - Necessary conditions for convergence rates of regularizations of optimal control problems T2 - System Modeling and Optimization : 25th IFIP TC 7 Conference, CSMO 2011, Berlin, Germany, September 12-16, 2011, Revised Selected Papers N2 - We investigate the Tikhonov regularization of control constrained optimal control problems. We use a specialized source condition in combination with a condition on the active sets. In the case of high convergence rates, these conditions are necessary and sufficient. KW - optimal control problem, inequality constraints, Tikhonov regularization, source condition Y1 - 2012 UR - https://link.springer.com/chapter/10.1007/978-3-642-36062-6_15 SP - 145 EP - 154 ER - TY - GEN A1 - Wachsmuth, Daniel A1 - Wachsmuth, Gerd T1 - How not to discretize the control T2 - Special Issue:Joint 87th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM) and Deutsche Mathematiker‐Vereinigung (DMV) N2 - In this short note, we address the discretization of optimal control problems with higher order polynomials. We develop a necessary and sufficient condition to ensure that weak limits of discrete feasible controls are feasible for the original problem. We show by means of a simple counterexample that a naive discretization by higher order polynomials can lead to non-feasible limits of sequences of discrete solutions. Y1 - 2016 UR - https://arxiv.org/abs/1603.07099 U6 - https://doi.org/10.1002/pamm.201610385 VL - 16 IS - 1 SP - 793 EP - 794 CY - Braunschweig ER - TY - GEN A1 - Wachsmuth, Daniel A1 - Wachsmuth, Gerd T1 - A simple proof of the Baillon-Haddad theorem on open subsets of Hilbert spaces T2 - Journal of Convex Analysis Y1 - 2023 UR - https://www.heldermann.de/JCA/JCA30/JCA304/jca30059.htm VL - 30 IS - 4 SP - 1319 EP - 1328 ER - TY - GEN A1 - Wachsmuth, Daniel A1 - Wachsmuth, Gerd T1 - A simple proof of the Baillon-Haddad theorem on open subsets of Hilbert spaces T2 - arXiv Y1 - 2022 UR - https://arxiv.org/abs/2204.00282 SP - 1 EP - 11 ER - TY - GEN A1 - Wachsmuth, Daniel A1 - Wachsmuth, Gerd T1 - Second-order conditions for non-uniformly convex integrands: quadratic growth in L¹ T2 - Journal of Nonsmooth Analysis and Optimizaton Y1 - 2022 U6 - https://doi.org/10.46298/jnsao-2022-8733 SN - 2700-7448 VL - 3 SP - 1 EP - 36 ER - TY - GEN A1 - Wachsmuth, Daniel A1 - Wachsmuth, Gerd T1 - Second-order conditions for non-uniformly convex integrands: quadratic growth in L^1 T2 - arXiv N2 - We study no-gap second-order optimality conditions for a non-uniformly convex and non-smooth integral functional. The integral functional is extended to the space of measures. The obtained second-order derivatives contain integrals on lower-dimensional manifolds. The proofs utilize the convex pre-conjugate, which is an integral functional on the space of continuous functions. Application to non-smooth optimal control problems are given. KW - Second-order optimality conditions KW - twice epi-differentiability KW - bang-bang control KW - sparse control Y1 - 2021 UR - https://arxiv.org/pdf/2111.10238.pdf ER -